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Work area for Practice Final Exam questions

Can anyone fill in the blanks on the last problem (23) Professor Bell worked in class today? I follow up to the u(x,t) = 1/2(sin2x)(cos2t) + ...... Where did the 1/2(sin2x)(cos2t) come from?


Can someone explain the purpose of the infinite sum 1/n^2 in problem 30? I understand how to use the Parseval's identity, but that last term in the problem statement is really confusing me.


I think there is a mistake on the solution sheet for number 12. Shouldn't the Eigenvectors be (1,1) or (-1,-1)? I don't see how they can be (-1,1) when I use the (B,-A) or (-B,A) rule. Can someone clarify?


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